(x+y)dy/dx=x-y.
Let y=vx, dy/dx=v+xdv/dx.
Substitute for y:
(x+vx)dy/dx=x-vx.
Divide through by x and substitute for dy/dx:
(1+v)(v+xdv/dx)=1-v,
v+xdv/dx+v²+vxdv/dx=1-v,
v²+2v-1+x(v+1)dv/dx=0,
(v²+2v-1)/(v+1)=-xdv/dx.
The variables separate:
(v+1)dv/(v²+2v-1)=-dx/x.
Note that d(v²+2v-1)/dv=2(v+1), so integrate:
½ln(v²+2v-1)=ln|a/x|, where a is a constant.
ln(v²+2v-1)=2ln|a/x|=ln(a/x)².
Therefore, v²+2v-1=(a/x)².
But v=y/x, so:
y²/x²+2y/x-1=a²/x².
Multiply through by x²:
y²+2xy-x²=a². This is the equation of a “tilted” hyperbola:
